Diffusion of ions in the channel

Given the presence of multiple K+ions within the selectivity filter of KcsA, a number

of simulations looked at the spontaneous motions of different configurations of K+

ions and water molecules in the filter. Several MD simulations of more than 1 ns duration have been carried out, e.g., [12, 14, 103, 104]. It is interesting to compare these simulations to see which sites within the filter are most often occupied by the

ions. In the crystal structure the two K+ions within the filter occupied S1 and (S3

or S4). The recent 2.0 ˚A structure actually shows 7 different binding sites, with ion

density (at high K+concentration) in sites S1-S4 as well as two more sites somewhat

outside the filter on the extra cellular side and one ion in the cavity. Comparing the various simulations, the preferred sites when two ions are present in the filter are: (i) S2 and S4 [103]; (ii) S2 and S4 [1] (iii) S1 and S3 [53]; (iv) S2 and S4 [6]; and (v) S2 and S4 (with a 3rd ion at S0) [12]. Interestingly, two independent simulations predicted there was a favourable location for a potassium ion outside the filter, which was confirmed by the recent high-resolution crystal structure [13, 104]. As discussed below, several free energy calculations [6, 13, 23] have suggested that the difference

in free energy between K+ions at S2 and S4, and at S1 and S3 is quite low. This is

consistent with the high permeation rate of potassium ions.

In the multi-nanosecond simulations concerted motions of the K+ions in the fil-

ter were seen. This is illustrated inFigure 2.11,from which it can be seen that the

K-W-K (i.e., ion-water-ion) triplet moves in a concerted fashion [104]. This is di- rect evidence for concerted single-file motion within a K channel selectivity filter. Clearly this complicates attempts to simulate ion flow through K channels as a dif- fusion process. It is also significant that in most simulations [1, 12, 103, 104] small

(generally ca. 0.5 ˚A) changes in conformation of the backbone carbonyls occur. In

particular, a ‘flipping’ of the carbonyl of V76 is observed. This is important, as it indicates that the conformation of the selectivity filter is not static, but can undergo dynamic changes on a timescale comparable to that of passage of the ions through the filter. Indeed, at low potassium concentrations (3 mM), ions are seen in the crys- tal structure mainly at S1 and S4, with some deformation of the filter consistent with observations in MD simulations. This may complicate mean field approaches, which thus far do not take protein flexibility into account, to simulation of ion permeation through KcsA.

2.3.5.2 Energetics of permeation

A number of groups have used atomistic simulations to explore the energetics of

permeation of KcsA. Allen et al. have calculated free energy profiles for K+and

Na+ions in a somewhat simplified model of a K channel, based on a channel-shaped

hydrophobic pore onto which a model of the KcsA filter is grafted [2]. Their results broadly support the ‘rigid filter’ model of K channel selectivity (see below). How- ever, the sensitivity of the results to initial assumptions of the rigidity of the filter is a little unclear. In a subsequent paper the same authors [1] using a complete model of the protein (but omitting the surrounding bilayer) found that the free energy differ-

Figure 2.11

Trajectories (along the pore axis) of K+ions (thick black lines) and water molecules

(gray lines) for two simulations with different starting configurations in sites S1-S4: (A) KA13C; (B) KA02C; Note that, for clarity, not all water molecules within the filter are shown. The locations on z (pore axis) of the four sites (S1 to S4) defined by the geometric center of the 8 oxygen atoms are indicated by the thin black lines. At each point in time, the origin of the coordinate system is defined as the center of gravity of the 16 oxygen atoms that line the selectivity filter. The black arrow in B

indicates the time at which a K+ion enters the selectivity filter from the extracellular

other groups have calculated potentials of mean force for ions in the selectivity filter. ˚

Aqvist and Luzhkov [6] showed that occupancy of sites S2 and S4 of the filter (see

Figure 2.11)by two K+ions was more favourable (by ca. 2 kcal./mol) than occu- pancy of sites S1 and S3. Other configurations were of higher free energy. Thus, a

permeation model based on switching of pairs of K+ions between these two con-

figurations was proposed. Berneche and Roux used umbrella sampling to calculate a two-dimensional free energy map describing possible pathways for translocating ions and suggest a plausible mechanism involving correlated motions of at least 3 ions and water on a relatively flat energy landscape [13]. A third study, by Burykin et al., also calculated potentials of mean force using free energy perturbation [6]. As such calculations are becoming increasingly feasible on standard computers, it seems likely there will be significant progress in this area in the near future.

2.3.5.3 Selectivity

Why are potassium channels so selective for potassium over sodium? The key differ- ences between potassium and sodium appear to be only a small difference in radius and in polarizability. On the basis of the X-ray structure of KcsA it has been sug- gested that a ‘rigid’ selectivity filter provides stronger cation-oxygen interactions for

K+ions than for Na+ions. Thus, the energetic cost of dehydrating K+ions is repaid

by ion/protein interactions, while ion/protein interactions are too weak to balance the

cost of dehydrating Na+ions. Several simulations have tried to address this question,

and suggest the picture might be somewhat more complex.

The deciding factor for selectivity in channels is that of the free energy of perme- ation; i.e., how the free energy of the system varies as different species of ion pass through the channel. The potential energies at various points along the central pore axis, which are much easier to calculate than free energies, are a first approximation.

Even with this type of calculation a difference between K+and Na+ions can be

observed [16]. However, for a more quantitative description free energy calculations are needed. Such calculations can yield the difference between two species of ions at a particular location, in addition to the full potential of mean force for moving a particular type of ion (as in the previous section). Allen et al. calculated that the

free energy (for a K+→ Na+transformation) is positive within the filter region [1],

which means it is more favourable for a potassium ion to be in the filter than it is for a sodium ion. However, the exact figure arrived at was somewhat sensitive to

the nature of the restraints applied to the filter during the simulation. ˚Aqvist and

Luzhkov [6, 78] have performed more detailed free energy perturbation calculations. Their results also supported the ‘rigid filter’ model of K channel selectivity. How- ever, it should be noted that in all three of these simulation studies it is not clear that the filter had time to fully ‘relax’ around the different species of cation. Longer MD

simulations of KcsA with K+ions or with Na+ions in the filter suggest that the filter

may be able to alter its conformation such that Na+ions can bind tightly within (and

thus block) the filter. The geometry of interaction of Na+ions with the filter appears

to be different from the geometry of interaction of K+ions [104].

mouth of the filter suggest a degree of selectivity in terms of which ions enter the filter [52, 104]. It is clear that very careful simulations are required to obtain the correct balance of ion/water, ion/protein and protein deformation energies. There is experimental data for other cations, e.g., rubidium [81]. In principle these could be simulated too, but they require additional testing of parameters because they are not commonly used in biomolecular simulations.

Clearly, it is becoming possible to carry out detailed numerical studies on potas- sium channels. The simulation results are sensitive to dynamic structural details and depend on simulation lengths and model accuracy, which might explain some of the differences in results from different labs. The fact that dynamic structural changes appear important will probably cause problems with respect to the use of restrained models (i.e., those omitting a lipid bilayer) to calculate permeation energetics. If such models are to be used, then care must be taken as to the strength and nature of the restraints. It also means simulation lengths need to be carefully checked to ensure sufficient sampling.

2.3.5.4 Interactions with toxins

Although I have not considered simulations of homology models of potassium chan- nels in detail, I would like to emphasize a relatively new direction in simulations of potassium channels for which the full structure is not known. Potassium channels and related channels show strong binding to certain toxins, either small molecules or peptides. The voltage-gated Shaker channel and other eukaryotic voltage-gated channels interact strongly with scorpion toxins such as charybdotoxin and agitoxin. The channel and toxin form very specific complexes with dissociation constants in the nanomolar range. For this reason, these toxins as well as others for different channels have been used extensively to probe the functional properties of ion chan- nels. By combining site-specific mutations in the toxin and in the channel, structural information on the channel (as the structure of many toxins is known) can be inferred from cooperative effects of mutations on the binding constant [57]. The resulting in- formation is a form of low-resolution structural information on the channel as well as on the mode of interaction between the toxin and the channel. Now that there are several high-resolution structures of potassium channels, molecular modelling and simulation studies can be used to understand how these toxins bind and interfere with channel function. Several recent studies have constructed models of voltage- gated channels and their interactions with toxins, and one study used the double mutant data to at the same time refine the model of the ion channel using several molecular-dynamics based techniques.

Cui et al. used Brownian dynamics simulations to dock the scorpion toxin Lq2, a member of the charybdotoxin family, in a model of a voltage-gated potassium chan- nel [37]. Lq2 has the interesting property that it blocks three families of potassium channels (voltage gated, calcium activated and inward rectifying channels), so that it is likely to interact with a common set of amino acids in the ion channels. This study used all 25 NMR structures for the toxin and studied their interactions simply by generating trajectories of the two proteins, without internal degrees of freedom

in the proteins, and analyzing the results. The main result is a good suggestion for the mode of docking, given the homology model for the potassium channel. Similar studies have been carried out on related channels and toxins.

Eriksson and Roux recently used the experimental data on agitoxin-Shaker inter- actions to refine a homology model of Shaker, and at the same time to determine how the toxin binds to the channel [46]. This method is significantly more involved, and uses thermodynamic data from double-mutant cycles to restrain the modes of interactions and the possible models. Their main result is a model of Shaker and a detailed description of how the toxin interacts with Shaker, including an explanation for some ambiguous experimental data. Without going into specific details of the results, this is an interesting development: it opens a range of new experimental data for use in model building and model validation, as well as a range of new conduc- tance data that e.g., BD simulations should be able to reproduce when the effects of the toxin is incorporated in BD simulations for cases where the toxin does not block completely.

These are technical uses, of interest in the context of this review, but of course there are also more practical implications for drug design. Ion channels already are an important target for drugs, or an important target for drugs to avoid (to prevent side effects). Two interesting examples of the use of double mutant cycle analyses, homology modelling and docking, followed by synthesis of new peptides with higher specificity as predicted by the theoretical work can be found in the work of Kalman et al. on voltage gated channels from T-lymphocytes [67] and from Rauer et al. on voltage and calcium gated channels from the same cells [87].

2.4

Outlook

Progress in modelling and simulation of ion channels in the last 5 years has been phe- nomenal. I think this progress has been inspired by a number of factors, including the availability of crystal structures of physiology relevant ion channels, the obvi- ous relevance of ion channels for biomedical and pharmaceutical research, the (at first sight) comparatively simple function and basic science of ion channels, the de- velopment of efficient and sophisticated simulation and modelling software, and the rapid increase in computer power available to an increasing number of researchers. In spite of this progress, we are still short of being able to link microscopic atomistic structures to macroscopic properties of ion channels. Nonetheless, there are several reasons to be optimistic about future work in this direction.

Molecular dynamics simulations include all atomic detail and can deal with pro- tein flexibility and conformational changes. They have been successfully used in a large number of studies to simulate local changes in structure and diffusion of water and ions, as well as to calculate potentials of mean force for ions in channels that can be used for BD simulations or kinetic theories. MD simulations are limited in

time scale, system size, and the accuracy of the description of atomic interactions. The time scale that is accessible depends mostly on the speed of computers, soft- ware, and algorithmic improvements, all of which combine to allow simulations of several orders of magnitude longer than currently possible. The accuracy of cur- rent parameter sets might not be high enough to, for example, distinguish accurately between different cations. However, the potential function in Equation 2.1 is not a fundamental property of molecular dynamics, and much more complex functions could be used, including potential functions that incorporate essential electronic ef- fects. Other improvements might be developed, perhaps based on a combination with semi-microscopic models, to deal more accurately with transmembrane poten- tials and differences in concentrations.

Brownian dynamics simulation is currently the most feasible way to link an ion channel structure to macroscopic properties, but it requires a description of the free energy profile for ion permeation and does not take protein flexibility into account. The latter might or might not be important for physiologically relevant channels. A description of the free energy profile is not easy to obtain. In most applications so far, only electrostatic interactions (combined with a simple short-range potential) were taken into account, calculated from Poisson or Poisson-Boltzmann equations. Free energy profiles for permeation have been calculated from MD simulations but these do not yet appear to be accurate enough. Nonetheless, these problems should be surmountable.

The limitations and prospects of mean field models depend on what they are being used for. I am not optimistic about the use of mean field models in which both ions and water are represented implicitly but a protein structure is represented in atomic detail, because the transition between mean field and atomic detail in one system is very large, and occurs on very short length scales. In many or maybe most ion channels specific interactions with ions and water appear important. In more sim- plified models, in which the protein is also simplified, mean field models are very interesting. They can suggest basic mechanisms for properties like selectivity, inde- pendent of atomic detail. Solving mean field models computationally only requires a fraction of the computational effort of molecular dynamics or Brownian dynamics simulations.

Combining methods from both atomistic and coarse-grained levels, using infor- mation from more detailed methods in less detailed methods that are closer to ex- perimental data seems a promising approach to understanding the properties of ion channels in atomic detail. A start has been made in the last few years, with exciting first results. As methods are developed further and additional experimental informa- tion becomes available, simulations should be able to provide detailed insight into ion channel structure-function relationships.

Acknowledgments DPT is a Scholar of the Alberta Heritage Foundation for Medical Research. Research in his group is supported by grants from the Canadian Institutes for Health Research (CIHR), the National Research Council for Science and Engi- neering (NSERC), and the Protein Engineering Network of Centres of Excellence (PENCE).

ing discussions and projects, including a review paper in Quarterly Reviews of Bio-

physics (2001. 34(4): p. 473-561) on which parts of this chapter are based. I would

also like to thank Dr. Drew Woolley and Kindal Robertson for their involvement in the alamethicin and OmpF work, respectively.

References

[1] Allen T.W., Bliznyuk A., Rendell A.P., Kuyucak S., and Chung S.H. (2000).

The potassium channel: structure, selectivity and diffusion. J. Chem. Phys. 112: 8191-204.

[2] Allen T.W., Kuyucak S., and Chung S.H. (1999). Molecular dynamics study of

the KcsA potassium channel. Biophys. J. 77: 2502-16.

[3] Allen R., Melchionna S., and Hansen J.P. (2002). Intermittent permeation of

cylindrical nanopores by water. Physical Review Letters 89: art. no.-175502.

[4] Andersen O.S., Apell H.J., Bamberg E., Busath D.D., Koeppe R.E., et al.

(1999). Gramicidin channel controversy - the structure in a lipid environment.

Nature Structural Biology 6: 609-.

[5] Anderson D.G., Shirts R.B., Cross T.A., and Busath D.D. (2001). Noncontact

dipole effects on channel permeation. V. Computed potentials for fluorinated gramicidin. Biophys. J. 81: 1255-64.

[6] Aqvist J., and Luzhkov V. (2000). Ion permeation mechanism of the potassium

channel. Nature 404: 881-4.

[7] Aqvist J., and Warshel A. (1989). Energetics of ion permeation through mem-

brane channels. Solvation of Na+by gramicidin A. Biophys. J. 56: 171-82.

[8] Ashcroft F.M. (2000). Ion Channels and Disease. London: Academic Press.

[9] Bass R.B., Strop P., Barclay M., and Rees D.C. (2002). Crystal structure

of Escherichia coli MscS, a voltage-modulated and mechanosensitive chan- nel.Science 298: 1582-7.

[10] Beckstein O., Biggin P.C., and Sansom M.S.P. (2001). A hydrophobic gating mechanism for nanopores. Journal of Physical Chemistry B 105: 12902-5. [11] Bek S., and Jakobsson E. (1994). Brownian dynamics study of a multiply-

occupied cation channel: application to understanding permeation in potassium channels. Biophys. J. 66: 1028-38.

[12] Berneche S., and Roux B. (2000). Molecular dynamics of the KcsA K+chan-

nel in a bilayer membrane. Biophys. J. 78: 2900-17.

h

K+channel. Nature 414: 73-7.

[14] Berneche S, and Roux B. (2001). Mechanism of ions permeation in the KcsA potassium channel. Biophys. J. 80: 674.

[15] Biggin P.C., and Sansom M.S. (2002). Open-state models of a potassium chan- nel. Biophys. J. 83: 1867-76.

[16] Biggin P.C., Smith G.R., Shrivastava I., Choe S., and Sansom M.S. (2001). Potassium and sodium ions in a potassium channel studied by molecular dy- namics simulations. Biochim. Biophys. Acta 1510: 1-9.

[17] Bilston L.E., and Mylvaganam K. (2002). Molecular simulations of the large conductance mechanosensitive (MscL) channel under mechanical loading. FEBS

Lett. 512: 185-90.

[18] Boda D., Henderson D., and Busath D.D. (2002). Monte Carlo study of the se- lectivity of calcium channels: improved geometrical model. Molecular Physics 100: 2361-8.

[19] Borisenko V., Burns D.C., Zhang Z.H., and Woolley G.A. (2000). Optical switching of ion-dipole interactions in a gramicidin channel analogue. J. Am.

Chem. Soc. 122: 6364-70.

[20] Borisenko V., Sansom M.S., and Woolley G.A. (2000). Protonation of lysine residues inverts cation/anion selectivity in a model channel. Biophys. J. 78: 1335-48.

[21] Breed J., Biggin P.C., Kerr I.D., Smart O.S., and Sansom M.S. (1997). Alame- thicin channels - modelling via restrained molecular dynamics simulations.

Biochim. Biophys. Acta 1325: 235-49.

[22] Brejc K., van Dijk W.J., Klaassen R.V., Schuurmans M., van Der Oost J, et al. (2001). Crystal structure of an ACh-binding protein reveals the ligand-binding domain of nicotinic receptors. Nature 411: 269-76.

[23] Burykin A., Schutz C.N., Villa J., and Warshel A. (2002). Simulations of ion current in realistic models of ion channels: The KcsA potassium channel.

Proteins-Structure Function and Genetics 47: 265-80.

[24] Cardenas A.E., Coalson R.D., and Kurnikova M.G. (2000). Three-dimensional Poisson-Nernst-Planck theory studies: influence of membrane electrostatics on gramicidin A channel conductance. Biophys. J. 79: 80-93.

[25] Capener C.E., Kim H.J., Arinaminpathy Y., and Sansom M.S.. (2002). Ion channels: structural bioinformatics and modelling. Hum. Mol. Genet. 11: 2425-33.

[26] Capener C.E., and Sansom M.S. (2002). Molecular dynamics simulations of a K channel model: Sensitivity to changes in ions, waters, and membrane envi- ronment. Journal of Physical Chemistry B 106: 4543-51.

tive of ab initio molecular dynamics in the study of biological systems. Acc.

Chem. Res. 35: 455-64.

[28] Cevc G. (1990). Membrane electrostatics. Biochim Biophys Acta 1031: 311- 82.

[29] Chang G., Spencer R.H., Lee A.T., Barclay M.T., and Rees D.C. (1998). Struc- ture of the MscL homolog from Mycobacterium tuberculosis: a gated mechano- sensitive ion channel. Science 282: 2220-6.

[30] Chung S.H., Allen T.W., and Kuyucak S. (2002). Conducting-state proper- ties of the KcsA potassium channel from molecular and Brownian dynamics simulations. Biophys. J. 82: 628-45.

[31] Chung S.H., Hoyles M., Allen T., and Kuyucak S. (1998). Study of ionic cur-